An IV administers medication to a patient's bloodstream at a rate of $3$ cubic centimeters per hour. At the same time, the patient's organs remove the medication from the patient's bloodstream at a rate proportional to the current volume $V$ of medication in the bloodstream. Which equation describes this relationship? Choose 1 answer: Choose 1 answer: (Choice A) A $\dfrac{dV}{dt}=-3kV$ (Choice B) B $\dfrac{dV}{dt}=3k-V$ (Choice C) C $\dfrac{dV}{dt}=3-kV$ (Choice D) D $\dfrac{dV}{dt}=k-3V$
Solution: The volume of medication in the bloodstream is denoted by $V$. The rate of change of the medication is represented by $V'(t)$, or $\dfrac{dV}{dt}$. Saying that the rate of change is proportional to something means it's equal to some constant $k$ multiplied by that thing. That thing, in our case, is the current volume, $V$, of medication in the bloodstream. We use a negative coefficient here to show that the change is removing medication, so we have $-kV$. Also, each hour the volume increases by a constant $3$ cubic centimeters, so we add that to the expression for the rate of change. In conclusion, the equation that describes this relationship is $\dfrac{dV}{dt}=3-kV$.